Basics of Differential Equations: Learn It 1

  • Determine the order of a differential equation
  • Tell the difference between a general solution and a particular solution
  • Identify what makes a problem an initial-value problem
  • Check if a function actually solves a given differential equation or initial-value problem

Basics of Differential Equations

Calculus is the mathematics of change, and we express rates of change using derivatives. This naturally leads us to one of calculus’s most powerful applications: differential equations.

A differential equation is simply an equation that contains an unknown function and its derivatives. These equations help us understand how quantities change over time and often reveal the underlying reasons for those changes.

Think of it this way: If you know how fast something is changing (the derivative), can you figure out what the original function was? That’s what differential equations help us solve.

Let’s start with a simple example: [latex]y' = 3x^2[/latex].

This equation tells us that we’re looking for a function [latex]y = f(x)[/latex] whose derivative equals [latex]3x^2[/latex]. In other words:

  • Start with some unknown function [latex]y = f(x)[/latex]
  • Take its derivative
  • The result must equal [latex]3x^2[/latex]

What function has a derivative equal to [latex]3x^2[/latex]? One answer is [latex]y = x^3[/latex], since [latex]\frac{d}{dx}[x^3] = 3x^2[/latex].

differential equations

Differential Equation: An equation involving an unknown function [latex]y = f(x)[/latex] and one or more of its derivatives.

[latex]\\[/latex]

Solution: A solution to a differential equation is a function [latex]y = f(x)[/latex] that satisfies the differential equation when the function and its derivatives are substituted into the equation.

Techniques for solving differential equations vary widely—from direct integration to graphical methods to computer calculations. We’ll explore the foundational ideas here and build on them throughout the course.

Here are some examples of differential equations and their solutions:

Differential Equation Solution
[latex]y' = 2x[/latex] [latex]y = x^2[/latex]
[latex]y' + 3y = 6x + 11[/latex] [latex]y = e^{-3x} + 2x + 3[/latex]
[latex]y'' - 3y' + 2y = 24e^{-2x}[/latex] [latex]y = 3e^x - 4e^{2x} + 2e^{-2x}[/latex]

Notice how these equations get more complex as we include higher-order derivatives like [latex]y''[/latex] (the second derivative).

CautionCaution solutions aren’t always unique! A differential equation can have multiple solutions! For example, [latex]y = x^2 + 4[/latex] is also a solution to [latex]y' = 2x[/latex] since the derivative of any constant is zero. We’ll explore this idea more shortly.

Before we dive deeper into what makes a function a solution, let’s review some key derivative rules you’ll need.

Derivatives of Exponential Functions

  1. [latex]\frac{d}{dx}(e^x) = e^x[/latex]
  2. [latex]\frac{d}{dx}(e^{g(x)}) = e^{g(x)} \cdot g'(x)[/latex] (chain rule with exponentials)

These rules will be essential as we work with differential equations involving exponential functions.

Verify that the function [latex]y={e}^{-3x}+2x+3[/latex] is a solution to the differential equation [latex]{y}^{\prime }+3y=6x+11[/latex].

Watch the following video to see the worked solution to the above example.

You can view the transcript for “4.1.2” here (opens in new window).